Thoughts on games (with probably too much computational complexity) updating every Tuesday and Friday some days during the school year

Friday, April 15, 2011

Game Description: FLex (Follow-the-Leader Hex)

After talking about Weak Hex (Whex)a few months ago, I tried playing it with some comrades here at Wittenberg. As a reminder, Whex is the version of Hex where both the first and second players must play adjacent to the first player's last move, with the winning condition exactly the same as before. Argimiro Quesada showed that this game is PSPACE-complete on general graphs, without needing to describe what happens if no adjacent move is possible (the graphs in his reduction are designed so that this never happens). When I asked him what the rule should be, he came up with the rule: if you can't play adjacent, then you lose.

This is a common first response for how to deal with this situation. While designing Atropos, I first considered using this same rule: if you can't play adjacent because there are no free spaces, you lose. Luckily, my advisor didn't like this and suggested the jumping: then you can play anywhere. These jumps greatly improve the game, and are also a big part of Adjex (Adjacent Hex).

While first sitting down with some other gamesters to play Whex, we tried the lose-when-no-adjacent-moves-

are-possible option, but found that to be a little unsatisfying. It is most exciting to have the game end when one player has built a path between their sides, not beforehand! We decided to add a jump rule here too.

Handling the jumps was a bit more tricky here, however, since the players have very different roles in Whex. Both players have to move adjacent to only the first player, not to whomever happened to play last. If the first player got to make a jump, then there is little question how to continue play. But what should you do if the second player gets to make the jump? Can they play anywhere they want? If so, are there any restrictions on where the first player goes afterwards? Can they play wherever they want, or do they have to play adjacent to where the second player just moved?

We came up with a nice solution for this case: switch the roles of the players. If the second player gets to jump, then they become the player who both have to play adjacent to instead of the first player.

Here are some more descriptive rules for the game, which we call Follow-the-Leader Hex (FLex):

This game is just like Hex, except at any time one player is the Leader and one is the Follower. On your turn, if possible you must play adjacent to the hexagon painted on the Leader's last move. If none are available, you may play on any unpainted hexagons on the board, and then you become the Leader, while your opponent becomes the Follower.

This game plays very nicely. In the first two games played between myself and Ernie, I won the first one without losing the "Leader" status, then won the second one starting as the Follower, fighting to become the Leader, getting the jump (and becoming the Leader) and using that to win. Having said that, in other games, we have seen the Follower force a win, so it's not clear which role is better. In fact, lately Ernie has had great insight into this game and playing as the Follower, who appears to have some advantage! Most of the games we play nowadays are determined by which player can get a jump first.

While playing, it is vital to use the Pie Rule to avoid a very simple immediate win (see this post on Whex). Unlike regular Hex, in FLex it may be more clear whether the second player should invoke the Pie Rule to win... maybe.

The credit for this game is really due to Argimiro; FLex is a minor tweak on Whex for a situation he had not previously considered. The tweak is due to Ernie, Obed and myself.

Enjoy! I am especially interested in hearing comparisons between Adjex and FLex. Which is more fun to play?

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Combinatorial Game Man

artist: Molly Dannaher

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This blog is devoted to Combinatorial Game Theory! Combinatorial games are two-player games with no randomness, perfect information (no one has any hidden information) and no draws allowed, though sometimes topics stray into other types of games as well.

Please let me know if you are interested in either writing a guest post or suggesting a topic. I would like this to be more reflective of CGT as a whole and not restrict it to my take on things. Additionally, please help me fill out the table of game properties, either by suggesting games or letting me know about results!